抛物线层与滤过Kronecker模的模空间

A global description of the fine Simpson moduli space of 1-dimensional sheaves supported on plane quartics

A global description of the fine Simpson moduli spaces of $1$-dimensional sheaves supported on plane quartics is given: we describe the gluing of the Brill-Noether loci described by Drézet and Maican and show that the Simpson moduli space $M=M_{4m\pm 1}(\mathbb P_2)$ is a blow-down of a blow-up of a projective bundle over a smooth moduli space of Kronecker modules. An easy computation of the Poincaré polynomial of $M$ is presented.

Filtered F-crystals on Shimura varieties of abelian type

In this paper, we define and construct canonical filtered $F$-crystals with $G$-structure over the integral models for Shimura varieties of abelian type at hyperspecial level defined by Kisin. We check that these are related by $p$-adic comparison theorems to the usual lisse sheaves, and as an application we also use this to show that the Galois representations generated from the $p$-adic étale cohomology of Shimura varieties with nontrivial coefficient sheaves are crystalline, at least in the case of proper abelian type Shimura varieties.

Residues and filtered D-modules

For an embedding of sufficiently high degree of a smooth projective variety X into projective space, we use residues to define a filtered holonomic D-module (M, F) on the dual projective space. This gives a concrete description of the intermediate extension to a Hodge module on P of the variation of Hodge structure on the middle-dimensional cohomology of the hyperplane sections of X. We also establish many results about the sheaves F_k M, such as positivity, vanishing theorems, or reflexivity.

范畴循环同调与栈上过滤 $\mathcal{D}$-模：Koszul 对偶

Categorical cyclic homology and filtered $\mathcal{D}$-modules on stacks: Koszul duality

本文证明光滑栈上过滤 $\mathcal{D}$-模与其形式环路空间上的 $S^1$-等变 ind-coherent 层之间存在 Koszul 对偶等价，并描述其特殊纤维与一般纤维。

Graded duality for filtered D-modules

For a coherent filtered D-module we show that the dual of each graded piece over the structure sheaf is isomorphic to a certain graded piece of the ring-theoretic local cohomology complex of the graded quotient of the dual of the filtered D-module along the zero-section of the cotangent bundle. This follows from a similar assertion for coherent graded modules over a polynomial algebra over the structure sheaf. We also prove that the local cohomology sheaves can be calculated by using the higher direct images of the twists of the associated sheaf complex on the projective cotangent bundle. Thes…

完全忠实的 p 进 Riemann--Hilbert 函子：作用于滤过凝聚 D-模

A fully faithful p-adic Riemann--Hilbert functor for filtered coherent D-modules

本文证明光滑刚性解析簇上滤过凝聚 D-模的求解函子完全忠实，恢复 Scholze 水平截面函子，并引入 de Rham Zariski 可构造层以推广经典 de Rham 局部系统。

Derived moduli of complexes and derived Grassmannians

In the first part of this paper we construct a model structure for the category of filtered cochain complexes of modules over some commutative ring $R$ and explain how the classical Rees construction relates this to the usual projective model structure over cochain complexes. The second part of the paper is devoted to the study of derived moduli of sheaves: we give a new proof of the representability of the derived stack of perfect complexes over a proper scheme and then use the new model structure for filtered complexes to tackle moduli of filtered derived modules. As an application, we const…

Harder-Narasimhan滤过的导出模栈的显式构造

An explicit construction of derived moduli stacks of Harder-Narasimhan filtrations

本文借助Behrend等人的方法，显式构造了特征0代数闭域上连通射影概形的Harder-Narasimhan滤过导出模栈，还研究了带滤过层的导出形变理论并与Di Natale的构造做了对比。

A noncommutative version of Beilinson's Theorem

We prove that the category of representations of the N-Kronecker quiver and that of coherent sheaves on the noncommutative projective scheme of $R=k< X_1,...,X_N >/(\sum^N_{i=1}X_i^2)$ are derived equivalent. This equivalence is easily proved by applying Orlov's Theorem, on the other hand,in our proof,the quadratic relation $\sum_{i=1}^NX_i^2$ naturally arises from Auslander-Reiten Theory.

mild difference modules的Stokes结构

Stokes structure of mild difference modules

本文通过滤子sheaves描述线性差分方程的Stokes现象，提出mild difference版本的Riemann-Hilbert对应，研究单复变量meromorphic连接的germs。