On $\mathsf{G}$-isoshtukas over function fields

Representations of the Kottwitz gerbes

Let $F$ be a local or global field and let $G$ be a linear algebraic group over $F$. We study Tannakian categories of representations of the Kottwitz gerbes $\text{Rep}(\text{Kt}_{F})$ and the functor $G\mapsto B(F, G)$ defined by Kottwitz in [28]. In particular, we show that if $F$ is a function field of a curve over $\mathbb{F}_q$, then $\text{Rep}(\text{Kt}_F)$ is equivalent to the category of Drinfeld isoshtukas. In the case of number fields, we establish the existence of various fiber functors on $\text{Rep}(\text{Kt}_{\mathbb{Q}})$ and its subcategories and show that Scholze's conjecture…

相对设置下Langlands参数的Jacobson–Morozov态射

The Jacobson--Morozov morphism for Langlands parameters in the relative setting

本文在相对设置下构造了SL₂-参数模空间与Jacobson–Morozov态射，证明其几何性质良好，并建立了与Weil–Deligne参数模空间在稠密开集、离散轨迹及Frobenius半单等价类层面的同构与双射关系。

Further remarks on derived categories of algebraic stacks

Let $X$ be an algebraic stack with quasi-affine diagonal of finite type over a field $k$ of characteristic $0$. We extend the well-known equivalence $\mathsf{D}^+(\mathsf{QCoh}(X)) \simeq \mathsf{D}_{\mathrm{qc}}^+(X)$ to unbounded derived categories. We also prove that if $X$ is smooth over $k$, then $\mathsf{D}_{\mathrm{qc}}(X)$ is compactly generated. We accomplish the former using the descendable algebras of Mathew. We also establish related results in positive and mixed characteristics.

Semi-abelian spectral data for singular fibres of the $\mathsf{SL}(2,\mathbb{C})$-Hitchin system

We describe spectral data for singular fibres of the $\mathsf{SL}(2,\mathbb{C})$-Hitchin fibration with irreducible and reduced spectral curve. Using Hecke transformations we give a stratification of these singular spaces by fibre bundles over Prym varieties. By analysing the parameter spaces of Hecke transformations this describes the singular Hitchin fibres as compactifications of abelian group bundles over abelian torsors. We prove that a large class of singular fibres are themselves fibre bundles over Prym varieties. As applications we study irreducible components of singular Hitchin fibre…

Rationality of Mukai varieties over non-closed fields

We discuss birational properties of Mukai varieties, i.e., of higher-dimensional analogues of prime Fano threefolds of genus $g \in \{7,8,9,10\}$ over an arbitrary field $\mathsf{k}$ of zero characteristic. In the case of dimension $n \ge 4$ we prove that these varieties are $\mathsf{k}$-rational if and only if they have a $\mathsf{k}$-point except for the case of genus $9$, where we assume $n \ge 5$. Furthermore, we prove that Mukai varieties of genus $g \in \{7,8,9,10\}$ and dimension $n \ge 5$ contain cylinders if they have a $\mathsf{k}$-point. Finally, we prove that the embedding $X \hook…

$\mathsf{Hilb}^n(\mathbb{C}^2)$ 的 Gromov-Witten 理论与 $\mathcal{A}_g$ 的 Noether-Lefschetz 理论

Gromov-Witten theory of $\mathsf{Hilb}^n(\mathbb{C}^2)$ and Noether-Lefschetz theory of $\mathcal{A}_g$

本文研究平面点Hilbert概形的亏格1 GW理论，借助与模空间上族局部曲线GW理论的对应计算基本一点不变量，发现其与主极化Abel簇模空间的NL理论平行结果一致，证明闭链类同态性质，并在非退化猜想下确定完整理论。