重新审视混合几何
Revisiting mixed geometry
AI总结：
本文提出了一种适用于任意Artin堆栈的统一构造方法，用于构建混合版本或分级提升，解决了Frobenius非半单性问题，并展示了其在几何表示论和结不变量中的应用。
摘要：
我们提供了一种统一的构造方法，用于构造“混合版本”或“分级提升”，这是在Beilinson-Ginzburg-Soergel的意义下提出的，适用于任意的Artin堆栈。特别是，我们得到了许多出现在几何表示论和分类化结不变量中的范畴的分级提升的一般构造。我们的新理论将每个有限型Artin堆栈$\mathcal{Y}$ over $\overline{\mathbb{F}}_q$关联到一个对称单向的DG-范畴$\mathsf{Shv}_{\mathsf{gr}, c}(\mathcal{Y})$，该范畴是$\mathcal{Y}$上的可构造分级sheaves，以及六 functor 形式主义，一个 perverse $t$-结构，以及在Bondarko和Pauksztello意义下的重量（或co-$t$-）结构，这些结构与六 functor 形式主义、pervasive $t$-结构和Frobenius权重在$\ell$-adic sheaves的范畴中兼容。经典上，混合版本只在非常特殊的情况下被构造，由于Frobenius的非半单性。我们的构造绕过了这个问题，通过半单化Frobenius作用本身。然而，$\mathsf{Shv}_{\mathsf{gr}, c}(\mathcal{Y})$在这些之前构造可用时与它们一致。例如，对于任何具有固定对$T\subset B$的最大torus和Borel子群的reductive群$G$，我们有monoidal DG重量范畴的等价$\mathsf{Shv}_{\mathsf{gr}, c}(B\backslash G/B) \simeq \mathsf{Ch}^b(\mathsf{SBim}_W)$，其中$\mathsf{Ch}^b(\mathsf{SBim}_W)$是monoidal $\mathsf{DG}$-category of bounded chain complexes of Soergel bimodules，$W$是$G$的Weyl群。
英文摘要：
We provide a uniform construction of "mixed versions" or "graded lifts" in the sense of Beilinson-Ginzburg-Soergel which works for arbitrary Artin stacks. In particular, we obtain a general construction of graded lifts of many categories arising in geometric representation theory and categorified knot invariants. Our new theory associates to each Artin stack of finite type $\mathcal{Y}$ over $\overline{\mathbb{F}}_q$ a symmetric monoidal DG-category $\mathsf{Shv}_{\mathsf{gr}, c}(\mathcal{Y})$ of constructible graded sheaves on $\mathcal{Y}$ along with the six-functor formalism, a perverse $t$-structure, and a weight (or co-$t$-)structure in the sense of Bondarko and Pauksztello, compatible with the six-functor formalism, perverse $t$-structures, and Frobenius weights on the category of (mixed) $\ell$-adic sheaves. Classically, mixed versions were only constructed in very special cases due to the non-semisimplicity of Frobenius. Our construction sidesteps this issue by semi-simplifying the Frobenius action itself. However, the category $\mathsf{Shv}_{\mathsf{gr}, c}(\mathcal{Y})$ agrees with those previously constructed when they are available. For example, for any reductive group $G$ with a fixed pair $T\subset B$ of a maximal torus and a Borel subgroup, we have an equivalence of monoidal DG weight categories $\mathsf{Shv}_{\mathsf{gr}, c}(B\backslash G/B) \simeq \mathsf{Ch}^b(\mathsf{SBim}_W)$, where $\mathsf{Ch}^b(\mathsf{SBim}_W)$ is the monoidal $\mathsf{DG}$-category of bounded chain complexes of Soergel bimodules and $W$ is the Weyl group of $G$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
